DYNAMIC AIRLINE PRICING AND SEAT AVAILABILITY. Kevin R. Williams. August 2017 Updated February 2018 COWLES FOUNDATION DISCUSSION PAPER NO.

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1 DYNAMIC AIRLINE PRICING AND SEAT AVAILABILITY By Kevin R. Williams August 2017 Updated February 2018 COWLES FOUNDATION DISCUSSION PAPER NO. 3003U COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box New Haven, Connecticut

2 DYNAMIC AIRLINE PRICING AND SEAT AVAILABILITY Kevin R. Williams School of Management Yale University February 2018 Abstract Airfares are determined by both intertemporal price discrimination and dynamic adjustment to stochastic demand. I estimate a model of dynamic airline pricing accounting for both forces with new flight-level data. With model estimates, I disentangle key interactions between the arrival pattern of consumer types and remaining capacity under stochastic demand. I show that the forces are complements in airline markets and lead to significantly higher revenues, as well as increased consumer surplus, compared to a more restrictive pricing regime. Finally, I show that abstracting from stochastic demand leads to a systematic bias in estimating demand elasticities. JEL: L11, L12, L93 kevin.williams@yale.edu This paper is an updated chapter of my dissertation at the University of Minnesota. I thank Tom Holmes, Jim Dana, Aniko Öry, Amil Petrin, Tom Quan and Joel Waldfogel for comments. I thank the seminar participants at the Federal Reserve Bank of Minneapolis, Yale School of Management, University of Chicago - Booth, Georgetown University, University of British Columbia, University of Rochester - Simon, Dartmouth University, Northwestern University - Kellogg, Federal Reserve Board, Federal Reserve Bank of Richmond, Indiana University, Indiana University - Kelley, Marketing Science, and the Stanford Institute for Theoretical Economics (SITE). I also thank the Minnesota Supercomputing Institute (MSI) for providing computational resources.

3 1 Introduction Air Asia (2013) on intertemporal price discrimination: Want cheap fares, book early. If you book your tickets late, chances are you are desperate to fly and therefore don t mind paying a little more. 1 easyjet (2003) on dynamic adjustment to stochastic demand: Our booking system continually reviews bookings for all future flights and tries to predict how popular each flight is likely to be. If the rate at which seats are selling is higher than normal, then the price would go up. This way we avoid the undesirable situation of selling out popular flights months in advance. 2 Airlines tend to charge high prices to passengers who search for tickets close to their date of travel. The conventional view is that these are business travelers and that airlines capture their high willingness to pay through intertemporal price discrimination. Airlines also adjust prices on a day-to-day basis, as capacity is limited and the future demand for any given flight is uncertain. They may adjust fares upward to avoid selling out flights in advance, or fares may actually fall from one day to the next, after a sequence of low demand realizations. This paper examines pricing in the airline industry, taking into account both forces: intertemporal price discrimination (fares responding to time) and dynamic adjustment to stochastic demand (fares responding to seats sold). Collectively, I call this dynamic airline pricing. I use a new flight-level data set consisting of daily fares and seat availabilities to estimate a model of dynamic airline pricing in which firms face a stochastic arrival of consumers. The mix of consumer types business and leisure travelers is allowed to change over time, and in the estimated model, late-arriving consumers are significantly more price-inelastic than early-arriving consumers. I use the model to quantify the welfare effects of dynamic airline pricing, but also to establish important interactions between intertemporal price discrimination and dynamic adjustment to stochastic demand in airline 1 Accessed through AirAsia.com s Investor Relations page entitled, What is low cost?" 2 Appeared on easyjet.com s FAQs. Accessed through Low-Cost Carriers and Low Fares" and Online Marketing: A Customer-Led Approach." 1

4 markets. Finally, I show that analyzing both forces together is critical to quantifying the welfare implications of airline pricing. The existing research documents the importance of intertemporal price discrimination and dynamic adjustment to stochastic demand separately in airline markets, and the central contribution of this paper is to study them jointly and quantify their interactions. Consistent with the idea of market segmentation, Puller, Sengupta, and Wiggins (2015) find that ticket characteristics, such as advance purchase discount (APD) requirements, explain much of the dispersion in fares. Lazarev (2013) quantifies the welfare effects of airline pricing by estimating a model of intertemporal price discrimination but not dynamic adjustment. He finds a substantial role for this force. 3 Escobari (2012) and Alderighi, Nicolini, and Piga (2015) find evidence that airlines face stochastic demand and that prices respond to remaining capacity. These results support the theoretical predictions of Gallego and Van Ryzin (1994) and a large branch of research in operations management that studies optimal pricing under uncertain demand, limited capacity, and limited time to sell. 4 Indeed, this research has been used to inform airline pricing algorithms. The estimated structural model allows me establish two key points about the interaction between the pricing forces. First, dynamic adjustment complements intertemporal price discrimination in the airline industry. This is due to the fact that price-inelastic consumers (i.e., business travelers) tend to buy tickets close to the departure date. To be able to price discriminate towards these late-arriving consumers, airlines must successfully save seats until close to their departure date. These consumers are then charged high prices. I show that dynamic pricing more efficiently allocates capacity and increases consumer welfare in the monopoly markets I study relative to either uniform pricing or a pricing 3 Intertemporal price discrimination can be found in many markets, including video games (Nair 2007), Broadway theater (Leslie 2004), and concerts (Courty and Pagliero 2012). Lambrecht et. al. (2012) provide an overview of empirical work on price discrimination more broadly. 4 An overview of the dynamic pricing literature can be found in Elmaghraby and Keskinocak (2003) and Talluri and Van Ryzin (2005). Sweeting (2012) analyzes ticket resale markets. Pashigian and Bowen (1991) and Soysal and Krishnamurthi (2012) study clearance sales and seasonal goods, respectively. Zhao and Zheng (2000), Su (2007), and Dilme and Li (2016) discuss extensions to dynamic pricing models, including consumer dynamics. Seat inventory control has also been studied; see Dana (1999). 2

5 system in which fares vary with only the purchase time but not with past demand shocks. Second, I show that in order to quantify the welfare effects of price discrimination in airline markets, it is necessary to take stochastic demand into account. When abstracting from stochastic demand, the opportunity cost of selling a seat is the same regardless of the date of purchase. In reality, the opportunity costs tend to increase in airline markets because of the late arrival of price insensitive consumers. As a consequence, I show an empirical procedure abstracting from this feature of the market will systematically overstate consumers price insensitivity. The bias is large close to the departure date. In order to investigate dynamic airline pricing, a detailed data set of ticket purchases is required. However, the standard airline data sets used in economic studies (e.g., Goolsbee and Syverson (2008); Gerardi and Shapiro (2009); Berry and Jia (2010)) are either at the monthly or the quarterly level. Recent papers use new data to get to the flight level. McAfee and Te Velde (2006) and Lazarev (2013) create data sets containing high-frequency fares. Other papers have obtained high frequency fares as well as a measure of seats sold. Puller, Sengupta, and Wiggins (2015) capture a fraction of sales through a single computer reservation system used by airlines. Escobari (2012) and Clark and Vincent (2012) collect fare and flight availability data, with the available number of seats derived from publicly available seat maps. I use a new data source that allows me to capture the same information that travel agents see, such as which seats are blocked, occupied, and available. I merge these data with daily fares and flight availability data. In total, I track over 1,300 flights in US monopoly markets over a six-month period. I use the data to provide descriptive evidence of both intertemporal price discrimination and dynamic adjustment to stochastic demand. While empirical evidence is informative, it cannot be used to disentangle the interactions between the pricing forces. I proceed by estimating a structural model that contains three key ingredients: (i) a monopolist has fixed capacity and finite time to sell; (ii) the firm faces a stochastic arrival of consumers; and (iii) the mix of consumers, correspond- 3

6 ing to business and leisure travelers, is allowed to change over time. 5 The firm solves a stochastic dynamic programming problem, offering a fare to consumers each day before departure. For the demand system, I assume that a stochastic process brings consumers to the market. The consumers that arrive know when they want to travel and solve a static discrete choice problem. 6 The demand model differs from earlier theoretical work, including Gale and Holmes (1993), and from empirical work such as Lazarev (2013), in which consumers are uncertain, and waiting provides more information. In my model the only reason to wait is to bet on price, and since prices tend to increase, I show that only a small transaction cost is needed to persuade consumers to decide whether to travel in the period they arrive. I also provide empirical evidence that suggests this is a reasonable assumption. 7 I use the firm s pricing decision, along with the realizations of demand, to separately identify the arrival process of consumers and the demand elasticity over time. Using the model estimates, I conduct a series of counterfactual exercises to quantify the welfare implications of dynamic airline pricing and to highlight how these results depend on the arrival process of consumers. The general finding is that, while more-restrictive pricing systems reduce the ability of firms to price discriminate, the aggregate consumer welfare gains are limited, or even negative, due to inefficient capacity allocation. That is, while airlines do price discriminate, which we may think only harms consumers, their pricing also more efficiently allocates scarce seats. In fact, under uniform pricing, revenues drop by 2%, but aggregate consumer welfare is unchanged. In this counterfactual, leisure consumers are charged relatively higher prices, and while business consumers are charged relatively lower prices, many flights sell out in advance. Moreover, I find total consumer welfare is lowered when fares are allowed to respond to the date of purchase but not to the scarcity of seats. However, the pricing forces interact differently under alternative arrival 5 McAfee and Te Velde (2006) note that stochastic demand models that do not incorporate changes in willingness to pay over time do not match the positive trend in airfares as the departure date approaches. 6 The specification is similar to that in Talluri and Van Ryzin (2004) and Vulcano, van Ryzin, and Chaar (2010); however, my model also allows for the mix of consumers to change across time. 7 Li, Granados, and Netessine (2014) study dynamic consumer behavior in airline markets. Depending on the specification, they find that between 5% and 20% of consumers are dynamic. 4

7 processes. If price-insensitive consumers were the first to arrive, for example, dynamic adjustment no longer complements intertemporal price discrimination, and stochastic demand pricing is used to clear excess capacity. This is consistent with fire sales that are commonly used in retailing. 2 Data I create an original data set of airfares and seat availabilities with data collected from two popular online travel services. The first is a travel search engine, owned by Kayak.com, from which I collect daily fares at the itinerary level, with itinerary defined as a routing, airline, flight number(s), and departure date(s) combination. I obtain all one-way and round-trip itinerary fares for stays of less than eight days. The fares recorded correspond to the cheapest ticket available for purchase. The fare data include the fare class and fare basis code, which provide some information regarding the restrictions of the ticket, such as advance purchase discount requirements. 8 The second web service, operated by Expertflyer.com, allows users to look up flight availability. I collect two sources of information. First, I obtain airline seat maps for each flight. By comparing seat maps across time, I obtain daily bookings. Second, I collect the fare availability (sometimes called fare buckets) for each flight. This information provides censored information regarding which fares are available to purchase. For example, Y9 indicates that the Y class fare, which is typically the most expensive coach fare available, has at least nine tickets available for purchase. 9 In total, the sample contains 1,362 flight departures, with each flight tracked for 60 days. All flights in my data set departed 8 G21JN5 is an example. This fare basis code indicates a G fare class ticket, and a 21-day advance purchase discount. Lazarev (2013) has a nice discussion of airline pricing and provides additional details on fare restrictions. Like Lazarev (2013), I model only the pricing choice and not the decision to assign restrictions or quantities for each price. 9 Regarding the example of the fare basis code G21JN5, the fare availability data might say G5. This means five seats remain for that particular fare. Fare availabilities can change because of ticket purchases, but also because of changes input by managers. Both of these potentially result in changes in price, which is the focus of this paper. 5

8 between March 2012 and September In the following subsections, I discuss route selection (Section 2.1) and the use of seat maps to infer bookings (Section 2.2), and I provide summary statistics and preliminary evidence from the data (Section 2.3). 2.1 Route Selection I select markets in which to study using the publicly available DB1B tables, which are frequently used to study airline markets. The DB1B tables contain 10% of domestic ticket purchases and are at the quarterly level. The data contain neither the date flown nor the purchase date. I define a market in the DB1B as an origin, destination, quarter. I single out markets where: (i) there is only one carrier operating; (ii) there is no nearby alternative airport; (iii) at least 95% of flight traffic is not connecting to other cities; (iv) total quarterly traffic is greater than 3,000 passengers; (v) total quarterly traffic is less than 30,000 passengers; and (vi) there is high nonstop traffic. Criteria (i) and (ii) narrow the focus to monopoly markets. One potential complication with using airline seat maps to recover bookings is that it is not clear what fare to assign to an observed change in a seat map. Since airlines offer extensive networks, the disappearance of a single seat could be associated with one of several thousand possible itineraries. This is an important consideration since pricing may be different across routes, and this is especially true with regard to feeder routes versus main routes. This issue is addressed in (iii), which essentially rules out most routes from major hubs. Criterion (iv) corresponds to routes with a less than 75% load factor (seats occupied / capacity) of a daily 50-seat aircraft. This criterion removes routes with irregular service. Criterion (v) 6

9 removes most routes with several daily frequencies. I then look for routes with (vi) high nonstop traffic. This criterion is important for establishing the relevant outside option in the demand model. In the data, (vi) is negatively correlated with distance (ρ =.5). Cities with very high nonstop traffic percentages tend to be short-distance flights, and, given such short distances, many consumers may choose to drive instead. At the same time, the DB1B data suggest that longer flights have lower nonstop traffic percentages, suggesting that more consumers choose a connecting flight option. I select four city pairs, or eight directional routings, given the selection criteria above. All directional routings either originate or end in Boston, MA. The other cities are: San Diego, CA; Austin, TX; Kansas City, MO; and Jacksonville, FL. The selected markets have close to 100% direct traffic, meaning that very few passengers connect to other cities. The percent of nonstop traffic ranges between 40% and 70%. Three of the four city pairs are operated by JetBlue and the other by Delta Air Lines. 10 low-frequency, with at most two daily frequencies. The selected markets are all Three other features of the data are worth noting. First, JetBlue prices itineraries at the segment level; that is, consumers wishing to purchase round-trip tickets on this carrier purchase two one-way tickets. As a consequence, round-trip fares in these markets are exactly equal to the sum of the corresponding one-way fares. Since fares must be attributed to each seat map change, this feature of the data makes it easier to justify the fare involved. Second, JetBlue does not oversell flights, while most other air carriers do. 11 I will use this feature of the data to simplify the pricing problem in the next section. Third, JetBlue does not offer first class on the routes studied. This allows for investigating all sales and also controls for one aspect of versioning (first class versus economy class) At the time of data collection, flights between Kansas City and Boston were operated by regional carriers on behalf of Delta Air Lines. Since Delta Air Lines determines the fares for this market, I collectively call these regional carriers Delta. 11 In the legal section of the JetBlue website, under Passenger Service Plan: JetBlue does not overbook flights. However some situations, such as flight cancellations and reaccommodation, might create a similar situation." 12 The Delta first class cabins were typically six seats for the market studied. 7

10 2.2 Inference and Accuracy of Seat Maps A seat map is a graphical representation of occupied and unoccupied seats for a given flight at a select point in time before the departure date. Many airlines that have assigned seating present seat maps to consumers during the booking process. When a consumer books a ticket and selects a seat, the seat map changes to reflect that an unoccupied seat is now occupied. The next consumer wishing to purchase a ticket on the flight is offered an updated seat map and can choose one of the remaining unoccupied seats. By differencing seat maps across time in this case daily inferences can be made about daily bookings. I use a new data source that allows me to distinguish between various types of occupied and occupied seats. Figure 1 presents a sample seat map: occupied seats are in solid blue, while the unshaded blocks correspond to unoccupied seats. Seats with a "P" are available, but classified as premium. These seats are located toward the front of the aircraft or in exit rows. Some airlines charge a premium to be seated in these rows. Finally, the seat map indicates seats currently blocked by the airline with "X"s. Seats that are blocked are usually not disclosed on airline websites; however, I am able to capture these data through the web service used. Seats may be blocked due to crew rest, weight and balance, because a seat is broken, or because the airline reserves accessible seats for handicapped passengers until the day of departure. 13 For every seat map collected, I aggregate the number of occupied, unoccupied, and blocked seats. I compare the aggregate counts across days to determine bookings by day prior to departure. 13 Seat blocking may be used to encourage consumers to purchase tickets or upgrade, as they give the impression that the cabin is closer to capacity. However, the data suggest that airlines predominantly block seats in exit rows and at the front and/or back of the cabin until closer to the departure date or when bookings demand additional seats. 69% of the flights in the sample experience changes in the number of blocked seats, and while I do not model the decision to block/unblock seats, I do take this information into account when determining bookings. Knowing which seats are blocked is important because it allows me to distinguish between consumers decisions and airlines adjustment of the supply of seats. For example, if an airline unblocks six seats, without these data, I would erroneously conclude that six passengers canceled their tickets. 8

11 Figure 1: An example seat map. The white blocks are unoccupied seats, the blue blocks are occupied, the blocks with the X s are blocked seats, and the blocks with a "P" are premium, unoccupied seats. Unfortunately, seat maps may not be accurate representations of true flight loads. This is especially problematic if consumers do not select seats at the time of booking. This measurement error would systematically understate sales early on, but then overstate last-minute sales when consumers without existing seat assignments are assigned seats. From a modeling perspective, this measurement error would lead to an overstatement 9

12 of the arrival of business consumers. Ideally, the severity of measurement error of my data could be assessed by matching changes in seat maps with bookings, however this is impossible with the publicly available data. While both imperfect, I perform two analyses to gauge the magnitude of the measurement error in using seat maps. First, I match monthly enplanements using my seat maps aggregated on the the day of departure with actual monthly enplanements reported in the T100 Segment tables. These tables record the total number of monthly enplanements by airline and route. Figure 2 provides a scatter plot that compares the two statistics. All the points closely follow the 45-degree line and I find seat maps differ with true enplanements by less than two percent on average. However, what really matters is the difference between actual loads and seat maps at the flight-day before departure level. Unfortunately, this analysis cannot be performed with the collected data set. Figure 2: Estimated Seat Map Measurement Error at the Monthly-Level Note: Measurement error estimated by comparing monthly enplanements using the T100 Tables and aggregating seat maps to the monthly level. The solid line reflects zero measurement error. The black-squares correspond to Delta observations. The two outliers are due to a change in flight numbers which were not picked up by the data collection process in one month. 10

13 Second, I create a new data set that allows me to estimate the measurement error at the flight-day before departure level. The mobile website version of United.com allows users to examine seat maps for upcoming flights. In addition, for premium cabins, the airline also reports the number of consumers booked into the cabin. I randomly select flights, departure dates, and search dates, and, in total, I obtain 15,567 observations. With these data I find that seat maps understate reported load factor by 2.3% on average (around 1-2 seats). Figure 3 plots the average measurement error by day before departure (t=60 corresponds to the day that flights leave), as well as a polynomial smooth of the data. I find the difference to range between 0% to 5% across days. Two caveats are worth noting. First, I cannot reject that the measurement error is constant across days. Seat maps are most accurate far in advance (60 to 40 days out) and close to the departure date. Second, while these data do not indicate all blocked seats and the main sample does, they are for a different airline, different markets, and for a different cabin. Figure 3: Estimated Seat Map Measurement Error by Day Before Departure Note: Measurement error estimated by comparing seat maps with reported load factor using the United Airlines mobile website. The dots correspond to the daily mean, and the line corresponds to fitted values of an orthogonal polynomial regression of the fourth degree. Total sample size is equal to 15,567 with an average load factor of 70.7%. 11

14 2.3 Summary Statistics and Preliminary Evidence Summary statistics for the data sample appear in Table 1. The average one-way ticket in my sample is $283, whereas the average round-trip fare is $528. The discrepancy in one-way and round-trip fares can be attributed to the flights operated by Delta since Delta does not price at the segment level, but JetBlue does. Table 1: Summary Statistics for the Data Sample Variable Mean Std. Dev. Median 5th pctile 95th pctile Oneway Fare ($) Load Factor Daily Booking Rate Daily Fare Change ($) Unique Fares (per itin.) Note: Summary statistics for 1,362 flights tracked between 3/2/2012 and 8/24/2012. The total number of observations is 79,856. Load Factor is reported between zero and one the day of departure. The daily booking rate and fare change differences across search days within flights. Finally, unique fares is at the itinerary level and denotes the number of unique price points per flight. Reported load factor is the number of occupied seats divided by capacity on the day flights leave, and is reported between 0 and 1. In my sample, the average load factor is 85%, ranging from 77% to 89% by market. The booking rate corresponds to the mean difference in occupied seats across consecutive days. I find the average booking rate to be 0.78 seats per day, per flight. At the 5th percentile, zero seats per flight are booked a day, and at the 95th percentile, four seats per flight are booked a day. Airline markets are associated with low daily demand, as 57% of the seat maps in the sample do not change across consecutive days. The fare change rate is an indicator variable equal to one if the itinerary fare changes across consecutive days. I find the daily rate of fare changes to be 20%, so that the itineraries in my sample typically change price 12 times in 60 days. On average, each itinerary reaches 6.8 unique fares, and given that the average itinerary sees 12

15 12 fare changes, on average, this implies that fares usually fluctuate up and down usually several times within 60 days. I use the institutional feature that fares are chosen from a discrete set (fare buckets) in the model. Figure 4: Frequency and Magnitude of Fare Changes by Day Before Departure Note: The top panel shows the percent of itineraries which see fares increase or decrease by day before departure. The lower panel plots the magnitude of the fare declines and increases by day before departure. The vertical lines correspond to advance-purchase discount periods (fare fences). Figure 4 shows the frequency and magnitude of fare changes across time. The top panel indicates the fraction of itineraries that experience fare hikes versus fare discounts by day before departure, and the bottom panel indicates the magnitude of these fare changes (i.e., a plot of first differences, conditional on the direction of the fare change). For example, in the left plot, 40 days prior to departure (t=20), roughly 10% of fares increase and 10% of fares decrease. The remaining 80% of fares are held constant. Moving to the bottom panel, the magnitude of fare increases and declines 40 days out is roughly $50. The top panel confirms fares change throughout time, and this is also true for fare declines. Note that well before the departure date, the number of fare hikes and declines is roughly even. The fraction of itineraries that experience fare hikes increases over time. 13

16 There are four noticeable jumps in the line, indicating fare hikes. These jumps correspond to crossing three, seven, 14 and 21 days prior to departure, or when the advance purchase discounts placed on many tickets expire. 14 The use of advance purchase discounts (APDs) is consistent with the story of intertemporal price discrimination. Surprisingly, though, the use of APDs is not universal. Only 20% of itineraries experience fare hikes at 21 days, and less than 50% increase at 14 days. Just under 70% of itineraries see an increase in fare when crossing the seven-day APD requirement. Figure 5: Mean Fare and Load Factor by Day Before Departure Note: Average fare and load factor by day before departure. Includes all oneway observations in the data sample. The vertical lines correspond to advance-purchase discount periods (fare fences). Moving to statistics in levels, Figure 5 plots the mean fare and mean load factor (seats occupied/capacity) by day before departure. The plot confirms that the overall trend in prices is positive, with fares increasing from roughly $225 to over $425 in sixty days. The noticeable jumps in the fare time series occur when crossing the APD fences noted in Figure 4. At 60 days prior to departure, roughly 40% of seats are already occupied. Consequently, I observe about half the bookings on any given flight. The booking curve 14 Advance purchase discounts are sometimes placed at four, ten, and 30 days prior to departure, but this is not the case for the data I collect. 14

17 for flights in the sample is smooth across time, leveling off at 85% roughly three days prior to departure. The fact that fares tend to double but that consumers still purchase tickets is suggestive evidence that consumers of different types purchase tickets towards the date of travel. Table 2: Dynamic Substitution Regressions (1) (2) (3) APD (0.042) (0.042) (0.042) APD (0.0548) (0.054) (0.054) APD (0.051) (0.050) (0.050) APD (0.066) (0.065) (0.065) m(t) Yes Yes Yes d.o.w. Flight FE No Yes d.o.w. Search FE No Yes Yes Flight FE No No Yes Observations 79,856 79,856 79,856 R Note: Flight clustered standard errors in parentheses. * p < 0.05, ** p < 0.01, *** p < m(t) is a third-order polynomial in days before departure, d.o.w. stands for day-of-week indicators for the day the flight leaves and the day of search. While Figure 5 shows that there are noticeable jumps in prices across time, it also shows there are no noticeable jumps in load factor over time. If consumers are aware that fares tend to increase sharply around APD fences, bunching would be expected in the booking curve. This is not the case, and formally testing for bunching (Table 2) reveals insignificant coefficients (except for the three-day APD). I use this finding to simplify the 15

18 demand model. That is, I assume that consumers make a one-shot decision to buy or not buy, and then revisit the idea of forward-looking behavior later. Finally, Figure 6 plots the mean fare response by day before departure. The graph separates out two scenarios: (1) situations in which the firm sees positive sales in the previous period; and (2) those in which there are no sales in the previous period. The graph shows that both pricing forces are at play. Conditional on positive sales, capacity becomes more scarce, and prices increase. This is consistent with stochastic demand pricing. Also consistent with stochastic demand pricing, prices decrease when sales do not occur, reflecting the declining opportunity cost of capacity. However, close to the departure date, regardless of sales, prices increase. This is inconsistent with a model of just stochastic demand and suggests that late-arriving consumers are less price-sensitive. Firms capture this higher willingness to pay through intertemporal price discrimination. Figure 6: Fare Response to Sales by Day Before Departure Note: Average fare changes as a response to sales by day before departure. The vertical lines correspond to advancepurchase discount periods (fare fences). The horizontal line indicates no fare response. 16

19 3 An Empirical Model of Dynamic Airline Pricing In this section, I specify a structural model of dynamic airline pricing. Section 3.1 provides an overview. Section 3.2 presents the demand model, and Section 3.3 presents the firm s problem. 3.1 Model Overview A monopolist airline offers a single flight for sale in series of sequential markets. I assume that the demand and pricing decisions are not correlated across departure dates (multiple flights), so investigating a single date is representative. Time is discrete and the airline pricing problem has a finite horizon. Period 1 corresponds to the first sales period, and period T corresponds to the day the flight leaves. Initial capacity is exogenous, and the firm is not allowed to oversell. Each period, the airline offers a single price to all customers. Consumers arrive according to a stochastic process. Each consumer is either a business traveler or a leisure traveler; business travelers are less price-sensitive than leisure travelers. The proportion of business versus leisure consumers is allowed to change across time. Upon arrival, consumers choose to purchase a ticket on the flight or choose not to travel, corresponding to the outside option. While the data (Section 2.3) suggest that modeling consumers as being myopic is a reasonable assumption, after estimating the model, I discuss the extension of allowing consumers to delay purchase. If demand exceeds remaining capacity, tickets are randomly rationed, which ensures that the capacity constraint is not violated. I also assume that passengers do not cancel tickets, as the average number of cancellations in the data per flight is fewer than two. Thus, remaining capacity is monotonically decreasing. With an updated capacity constraint, the firm again chooses a fare to offer, and the process repeats until the perishability date. 17

20 3.2 Demand Each day before the flight leaves, a Poisson process ( M t ) brings new consumers to the market. Upon entering the market, all uncertainty about travel preferences is resolved. This approach differs from Lazarev (2013) and earlier theoretical work, including Gale and Holmes (1993), in which existing consumer uncertainty can be resolved by delaying purchase. In this model, at date t, consumers arrive, and choose to either purchase a ticket or exit the market. The demand model is based on the two-consumer type discrete choice model of Berry, Carnall, and Spiller (2006), which is frequently applied to airline data. Consumer i is a business traveler with probability γ t or a leisure traveler with probability 1 γ t. Consumer i has preferences (β i, α i ) over product characteristics (x t R K ) and price (p t ), respectively. If consumer i chooses to fly, the consumer receives utility U it1 = x t β i α i p t + ε it1. If i chooses to not fly, the consumer receives normalized utility U it0 = ε it0. Assume that the idiosyncratic preferences of consumers, (ε it1, ε it0 ), are i.i.d. Type-1 Extreme Value (T1EV). Following the discrete choice literature, consumer i chooses to fly if and only if U it1 U it0. Define y t = ( α i, β i, ε i1, ε i0 to be the vector of preferences for the consumers that )i 1,.., M t enter the market. Suppressing the notation on product characteristics for the rest of this section, the demand for the flight at t is defined as M t Q t (p t, y t ) := 1 [ ] U it1 U it0 {0,..., M t }, i=0 where 1( ) denotes the indicator function. Demand is integer-valued; however, it may be the case that more consumers want to travel than there are seats remaining i.e., Q t (p, y) > s t, where s t is the number of seats remaining at t. Since the firm is not allowed to oversell, in these instances, I assume that remaining capacity is rationed by random selection. Specifically, I assume that the market first allows consumers to enter and choose the product that maximizes utility. After consumers make their decisions, the capacity 18

21 constraint is checked. If demand exceeds remaining capacity, consumers who wished to purchase are randomly shuffled. The first s t are selected, and the rest receive their outside option. Although the model assumes that consumers arrive and purchase a single one-way ticket, it allows for round-trip ticket purchases in the following way: a consumer arrives looking to travel, leaving on date d and returning on date d. The consumer receives idiosyncratic preference shocks for each of the available flights in both directions, and chooses which tickets to purchase. Since several airlines, such as JetBlue, price at the segment level, there is no measurement error in this procedure. That is, a consumer pays the same price for for two one-way tickets as the consumer would for a round-trip ticket. 15 The assumptions placed on the consumer problem, along with the assumption that the firm cannot oversell, allow for deriving closed-form expressions to the demand system. As the firm (and econometrician) does not know how many consumers, and how many consumers of each type will arrive, I integrate over the distribution of y t, Q e t (s t, p t ) = min ( ) Q t (p t, y t ), s t dft (y t ). y t To start, note that the T1EV assumption on the consumer preference shocks leads to the frequently used conditional logit model. Since there is only a single product in the choice set, π i t1 := Pr(i wants to purchase j type = i) = exp( x t β i + α i p t ). The discrete choice literature typically does not model capacity constraints. Because consumers may be forced to the outside option once the capacity constraint binds, the purchase probabilities represent desired purchases instead of realized purchases. Let π i t0 denote the purchase probability of the outside option for consumer type i. 15 Since remaining capacity is rationed when an oversell would otherwise occur, an implication of this modeling choice is that a consumer may be forced to the outside option for either the outbound or the inbound leg of the round-trip. 19

22 Let B denote the business type and L denote the leisure type. Recall that the probability of a consumer being type-b is γ t. Then, γ t π B t defines the probability that a consumer is of the business type and wants to purchase a ticket. Consider the market share of the outside good at time t. Conditional on k N consumers arriving (suppressing the dependence on p t, x t ), the probability that all customers want to purchase the outside option is Note that Pr ( Q t = 0 k ) = k n=0 ( ) k ((1 n ( ) k n. γt )π L n t0) γt π B 16 t0 Pr t ( Qt = 0 ) = Pr t (Q t = 0, M t = k) = k=0 Pr t (Q t = 0 M t = k)pr( M t = k), k=0 and since consumers are assumed to enter the market according to a Poisson process, 17 Pr ( Q t = 0 ) has the following analytic form: Pr t ( Qt = 0 ) = k=0 µ k t e µ t k! k n=0 ( ) k ((1 n ( k n. γt )π L n t0) γt πt0) B (3.1) The expression for selling a positive number of seats is similar, in that the expression goes through all combinations of choosing to fly, or choosing not to fly, and mixes over 16 For example, let k = 2. Then, conditional on two consumers arriving, both consumers want to purchase the outside option. Both consumers are leisure travelers; both consumers are business travelers; or one of each type arrive. The first two situations correspond to n = 2 and n = 0, respectively, since [(1 γ t )π L t0 ]2 is the probability of two leisure consumers arriving and wanting to choose the outside option, and [γ t ς B t0 ]2 is the probability two business consumers want to choose the outside option. Lastly, it could be the case that one business and one leisure consumer arrive. There are two possibilities: the first consumer is the business ( γπ B t0) enters the probability. consumer, or vice versa. Hence, 2 ( ) 1 γ t π L 17 t0 With this assumption, the demand model closely follows Talluri and Van Ryzin (2004) and Vulcano, van Ryzin, and Chaar (2010), except that this model has two consumer types. 20

23 the two consumer types. It also is a Poisson-Binomial mixture and can be shown to equal Pr t ( Qt = q 0 < q < s ) = k=q µ k t e µ t k! ( ) q ( ) k q (γt l ( q l π B q l t1) (1 γt )πt1) L (3.2) l=0 k q ( ) k q ((1 n ( γt )π L n t0) γt πt0) k q n B. n=0 Demand is latent in the case of a sellout since it is possible that some consumers are forced to the outside option. These probabilities can be constructed based on the fact that at least s t seats are demanded. It can be shown that the Binomial-Mixture is equal to Pr t ( Qt q s ) = q=s k=q µ k t e µ t k! ( ) q ( ) k q (γt l ( q l π B q l t1) (1 γt )πt1) L (3.3) l=0 k q ( ) k q ((1 n ( γt )π L n t0) γt πt0) k q n B. n=0 Thus, all the demand possibilities (Equation ), which will be denoted f t (s s, p), have analytic expressions. 3.3 Monopoly Pricing Problem The monopolist maximizes expected discounted revenues in a series of sequential markets. In each of the sequential markets, the airline chooses to offer a single price to all consumers. Because of the institutional feature that airfares are discrete, I assume that the firm chooses a price from a discrete set (P). The pricing decision is based on the states of the flight: seats remaining, time left to sell, flight characteristics x j (notation suppressed), as well as idiosyncratic shocks ω t := {ω tp : p t P} R P, which are assumed to be distributed i.i.d T1EV, with scale parameter σ. These shocks are assumed to be additively separable to the remainder of the per-period payoff function, which, in this case, is expected revenues, R e t (s, p) = p t Q e t (s t, p t ). 21

24 The firm s problem can be written as a dynamic discrete choice model. Let V t (s t, ω t ) be the value function given the state (t, s t, ω t ). Denoting δ as the discount factor, the dynamic program (DP) of the firm is V t (s t, ω t ) = max p t P ( ) R e t (s t, p t ) + ω tp + δ V t+1 (s t+1, ω t+1 )dh t (ω t+1, s t+1 s t, p t, ω t ). ω t+1,s t+1 s t,p t,ω t Because the firm cannot oversell, capacity transitions as s t+1 = s t min { Q t (p t, y y ), s t }. The firm faces two boundary conditions. The first is V t (0, ω t ) = 0 that is, once the airline hits the capacity constraint, it can no longer sell seats. The second is V T (s t, ω t ) = 0 that is, unsold seats on the day that the flight leaves represent lost revenue opportunities. I assume that conditional independence is satisfied, meaning that the transition probabilities can be written as h t (s t+1, ω t+1 s t, ω t, p t ) = g(ω t+1 ) f t (s t+1 s t, p t ). 18 (1987), the expected value function can be expressed as EV t (p t, s t ) = s t+1 p t+1 P Following Rust ( R e σ ln t+1 exp (s ) t+1, p t+1 ) + EV t+1 (p t+1, s t+1 ) σ f t(s t+1 s t, p t )ds t+1 +σφ, and the conditional choice probabilities also have a closed form and can be computed as CCP t (s t, p t ) = exp {( R e t (p t, s t ) + EV t (p t, s t ) ) /σ } p t P [ exp { R e t (p t, s t) + EV t (p t, s t) } /σ ]. Given a set of flights (F) each tracked for (T) periods, the likelihood for the data, accounting for the firm s pricing decision, is given by max θ L(data θ) = max CCP t (s t, p t ) f t (s t+1 s t, p t ), (3.4) θ F 18 Since common parameters enter both the payoff function and the transition probabilities, I do not estimate the transition probabilities in a first stage, such as in Hotz and Miller (1993) and Bajari, Benkard, and Levin (2007). T 22

25 where θ := ( β, α, γ, µ, σ ) are the parameters to be estimated. 4 Model Estimates In this section, I discuss the identification and the estimation procedure (Section 4.1) and then the results and model fit (Section 4.2). Finally, I discuss the model extension of allowing consumers to delay purchase (Section 4.3). 4.1 Identification and Estimation The key identification challenge of the paper is to separately identify the demand parameters from the arrival process. This is pointed out in Talluri and Van Ryzin (2004), for example. The issue arises because without search data to pin down the arrival process, an increase in arrivals could instead be seen as a change in the mix of consumer types (demand). For example, the sale of two seats could have occurred because two consumers arrived and both purchased or because four consumers arrived and half purchased. Note that this is an argument about the demand elasticity, and I argue that accounting for the firm s pricing decision allows me to disentangle the two. Figure 6 demonstrates why the pricing information is useful in separating the overall demand elasticity from the arrival process. Given stochastic demand, we would expect prices to rise when demand exceeds expected demand, and to fall over a sequence of low demand realizations. However, close to the departure date and regardless of sales, Figure 6 shows that prices tend to rise. This would only occur in the model presented if there is an overall change in the demand elasticity. That is, consumers who shop late are less price sensitive than those who shop early. By solving the firm s dynamic programming problem, I recover the opportunity costs of seats, and along with price, I recover the demand elasticity. Changes in average prices over time inform the demand elasticity, and responses to variation in sales given seats remaining and time left to sell inform the arrival process. 23

26 The identification strategy utilizes the firm s pricing decision, but this creates a significant computational burden in estimation as the finite-horizon, non-stationary firm problem needs to be solved for each candidate vector of parameters. Thus, while the model is general to include flight characteristics, such as day-of-week effects, holiday indicators, and potential several flight options 19, incorporating these features only increases the computational burden. To keep the problem tractable, I estimate the model by abstracting from flight characteristics other than price, but allow for city-pair-specific parameters. The model fits the pricing data well, as shown in the next section. I assign the discount factor to be one. Finally, I place additional restrictions on γ t and µ t, or the probability on consumer types and arrival rates, respectively. I assign γ t to be a logit function of two parameters. This forces monotonicity, but not strict monotonicity, in the mixture of business versus leisure customers over time. I assign: µ 1 Greater than 21 days prior to departure; µ 2 14 to 21 days prior to departure; µ 3 7 to 14 days prior to departure; and µ 4 Less than 7 days prior to departure which corresponds to the advance purchase discount periods commonly seen in airline markets. This adds some flexibility in the Poisson arrivals of customers. As consumer types are allowed to change daily, this allows for day-to-day variation in expected demand. Finally, I assume β L = β B, resulting in = 10 parameters to be estimated per city pair, or 40 parameters to be estimated in total. To estimate the problem, I maximize the log-likelihood of the firm s dynamic programming problem found in Figure 3.4 using the Knitro solver. I utilize 100 random starts in the parameter space. The state space is size T S Q P per route, which is the number of periods, times number of potential sales and current seats remaining, times number of prices in the choice seat. This is a large state space, even though many entries are zero due 19 Additional assumptions on the rationing rule are required in this case. 24

27 to restrictions implied by the model, i.e., monotonically decreasing capacity. To decrease the size of the problem, I cluster prices using k-means (size 8), and then take the mean of the clusters, for each market. This groups together fares that only differ by sometimes a dollar, and the resulting fares have a geometric fit of 93-98% with the raw data. This reduces the state space by a factor of 2-3, and results in a state space of up to a few million per city pair. Finally, I use an important institutional feature of airline markets to define the price choice sets of firms. Since airlines commonly assign fare restrictions, such as advance purchase discounts, this creates a restriction that some fares are only used at particular times. For example, a fare with a 21-day advance purchase discount (APD) requirement is not available for purchase close to the departure date. I use the fare restrictions (fences) attached to the fares to define daily choice sets, P t. Fares with early APD requirements are less expensive than fares that are offered close to the departure date. I demonstrate later how firm pricing varies across time without imposing the fences Model Estimates, Model Fit, and Model Pricing Parameter estimates appear in Table 3. All parameters are significant at the 1% level. The parameter estimates suggest leisure consumers are twice as price sensitive as business consumers on average, and business consumers are willing to pay 60% more in order to secure a seat, on average. 21 Demand elasticities range from two to four depending on market and time until departure. The probabilities on consumer types implies a significant change in the price sensitivity of consumers over time. Figure 7 plots the fitted values using the (γ 1, γ 2 ) parameters for the markets studied. The estimates imply early on that essentially all arrivals are leisure consumers. Then, a few weeks prior to departure, there is a steep shift towards more price inelastic arrivals. Moving to the parameters governing 20 Advance purchase discounts are modeled from the set of daily fares offered, after clustering. This accounts for the fact that high fares are offered close to the departure date and low fares are not. 21 The model is estimated where prices are measured in hundreds of dollars. 25